I am doing max/min problems in calculus. The problem is to maximize the volume of a rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid x^2/4 + y^2 + z^2/9 = 1

I´d try this: There is only one degree of freedom you have: The extension of the box in one of the axis-direction (because this extension automatically gives you the other two extensions, at least when you assume that the box touches the ellipsiod, which seems a sensible assumption). Try to express the volume of the box by this one parameter, then find the maximum as usual (by taking the derivative with respect to this parameter). I cannot guarantee you that it works because I haven´t worked it out, but it´s what would be my first attempt.

you have to specify at least two axis values for the other to fall into place.

if you specify one, you end up with an ellipse of possible solutions.

 

i think substitution is the way to go, im not sure how far you can take it, but i've taken it down to v = f(x,y)

potentially you could make a 3d graph and find the maximum value for the volume by taking lots of x,y combinations.

a little programming wouldn't go astray.

I solved the problem, thanks for the help. I ended up using Lagrange Multipliers.